In its simplest form the magnetoelastic buckling instability refers to the sudden bending transition of an elastic rod experiencing a uniform induction field applied at a normal angle with respect to its long axis. This fundamental physics phenomenon was initially documented in 1968, and, surprisingly, despite many refinements, a gap has always remained between the observations and the theoretical expectations. Here, we first renew the theory with a simple model based on the assumption that the magnetization follows the rod axis as soon as it bends. We demonstrate that the magnetoelastic buckling corresponds to a classical Landau second-order transition. Our model yields a solution for the critical field as well as the shape of the deformed rods which we compare with experiments on flexible ferromagnetic nickel rods at the centimeter scale. We also report this instability at the micrometer scale with specially designed rods made of nanoparticles. We characterized our samples by determining all of the relevant parameters (radius, length, Young modulus, magnetic susceptibility) and, using these values, we found that the theory fits extremely well the experimental results for both systems without any adjustable parameter. The superparamagnetic feature of the microrods also highlights the fact that ferromagnetic systems break the symmetry before the buckling. We propose a magnetic “stick–slip” model to explain this peculiar feature, which was visible in past reports but never detailed.

https://www.pnas.org/content/pnas/112/23/7135.full.pdf

A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods.

This study presents the buckling analysis of soft ferromagnetic FG circular plates made of poro material. Equilibrium and stability equations of a poro circular plate in transverse magnetic field are derived. This study analyzes the poroelastic instability of clamped edge ferromagnetic plates subjected to magnetic loadings. The geometrical nonlinearities are considered in the Love–Kirchhoff hypothesis sense. In this paper the effect of pore pressure on critical magnetic field of plate and the effect of important parameters of poroelastic material on buckling capacity are investigated. Also the compressibility of fluid and porosity on the buckling strength are being investigated.

Buckling analysis of thin circular FG plates made of saturated porous-soft ferromagnetic materials in transverse magnetic field

Active control of beam with magnetostrictive layer

Magnetorheological elastomers (MREs) are ferromagnetic particle impregnated rubbers whose mechanical properties are altered by the application of external magnetic fields. Due to their coupled magneto-mechanical response, MREs are finding an increasing number of engineering applications. One such application is in haptics, where the goal is to actively control surface roughness. One way to achieve this is by exploiting the unstable regime of MRE substrate/layer assemblies subjected to transverse magnetic fields. In this work, we study the response of such an assembly subjected to a transverse magnetic field and in-plane stress. The layer is made up of a transversely isotropic MRE material, whose energy density has been obtained experimentally, while the substrate is a non-magnetic isotropic pure polymer/gel. An analytical solution to this problem based on a general, finite strain, 2D continuum modeling for both the MRE layer and the substrate, shows that for adequately soft substrates there is a finite-wavelength buckling mode under a transverse magnetic field. Moreover, the critical magnetic field can be substantially reduced in the presence of a compressive stress of the assembly, thus opening the possibility for haptic applications operating under low magnetic fields.

https://hal-polytechnique.archives-ouvertes.fr/hal-00995717/document

Instability of a magnetoelastic layer resting on a non-magnetic substrate

Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value. This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors. The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block's aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0.25 ) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio.

Magnetoelastic buckling of a rectangular block in plane strain

Experimental and Theoretical Study on Magnetoelastic Buckling of a Ferromagnetic Cantilevered Beam-Plate

A method of numerical analysis of the magnetoelastic buckling of toroidal field coils is shown on the basis of finite element method. Lagrange’s equation is used to deduce an eigenvalue equation determining the buckling current of the toroidal coils. The Lagrangian consists of elastic strain energy and magnetic energy. Since the coil is unidirectional, fiber-reinforced composite material, the theory of orthotropy is applied to determine elastic constants at 4.2 K. A buckling experiment on the eight superconducting-coil full torus was conducted to verify the numerical method developed here. A comparison of the numerical results with the experimental result is fairly good.

K. Someya

Papers

Experimental and Theoretical Study on Magnetoelastic Buckling of a Ferromagnetic Cantilevered Beam-Plate

Finite Element Analysis of Magnetoelastic Buckling of Eight-Coil Superconducting Full Torus

Experimental study of energy release due to cracking of epoxy impregnated conductors

A new technique for detection of dynamic crack initiation